Stochastic methods for physics and biology, spring 2010
The aim of the course is to introduce the basic concepts of the theory of stochastic differential equations needed in applications (applied mathematics, physics and biology). In particular we will illustrate methods of qualitative, asymptotic and numerical analysis.
Weeks 3-9 and 11-18, Monday 14-16 in room C124 and Friday 14-16 in room C123.
Easter holiday 1.-7.4.
First lecture: Friday, 22.01.2010
The lecture notes cover and sometimes integrate the material expounded in the lections. They also give bibliographic references for the same topics. For exercises please refer to the section below.
Lecture_01: Background of Probability
Lecture_11: Kolmogorov-Chentsov theorem
Lecture_02 (rev 29.01): Bernoulli variables, Borel-Cantelli lemma
Lecture_03: Limit theorems
Lecture_13: Ito integral
Lecture_14: Ito and Stratonovich calculus
Lecture_05: Numerical generation of random variables
Lecture_15: SDE, existence and uniqueness
Lecture_06 (rev 12.02): Statistical tests
Lecture_16: Kolmogorov pair and diffusion
Lecture_07: Fokker-Planck equation I
Lecture_17: Exit time statistics
Lecture_08: Fokker-Planck equation II
Lecture_18: Girsanov formula
Lecture_09: Brownian motion
Lecture_19: Numerical integration of SDE's
Lecture_10: Karhunen-Loève representation
Lecture_20: Population dynamics
There will be two options for taking the course. Passing an exam to be held after the end of the course, in
date to be agreed or working out a short project (recommended).
- L.C. Evans, "An Introduction to Stochastic Differential Equations", lecture notes.
- P. E. Kloeden, E. Platen, H. Schurz, "Numerical Solution Of Sde Through Computer Experiments", Springer (Universitext) (2003).
- N. Berglund, B. Gentz "Noise-Induced Phenomena in Slow-Fast Dynamical Systems. A Sample-Paths Approach", Springer (Probability and its Applications) (2005).
- D.J. Higham, "An algorithmic introduction to numerical simulation of stochastic differential equations" SIAM Review, Education Section, 43, 2001, 525-546. (Link to Higham's publications page.)
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